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📦 version 3.12.0, in opensuse-leap 16.0

💖 CPORFSX — cporfsx.f(3) 💖

okay bestie, this is the cporfsx.f(3) page from the Library Functions Manual, and we're talking about the CPORFSX subroutine. She improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite, and she also provides error bounds and backward error estimates for the solution. The vibe is very “trust me, I checked the error bounds.” 💅

🌸 NAME 🌸

The NAME section is literally just this source path, bestie:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/cporfsx.f

✨ SYNOPSIS ✨

🦋 Functions/Subroutines 🦋

       subroutine CPORFSX (uplo, equed, n, nrhs, a, lda, af, ldaf, s, b, ldb,
           x, ldx, rcond, berr, n_err_bnds, err_bnds_norm, err_bnds_comp,
           nparams, params, work, rwork, info)
           CPORFSX

💖 Function/Subroutine Documentation 💖

Here's the full signature, exact and iconic:

   subroutine CPORFSX (character uplo, character equed, integer n, integer
       nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension(
       ldaf, * ) af, integer ldaf, real, dimension( * ) s, complex, dimension(
       ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx,
       real rcond, real, dimension( * ) berr, integer n_err_bnds, real,
       dimension( nrhs, * ) err_bnds_norm, real, dimension( nrhs, * )
       err_bnds_comp, integer nparams, real, dimension(*) params, complex,
       dimension( * ) work, real, dimension( * ) rwork, integer info)
       CPORFSX

✨ Purpose ✨

CPORFSX improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite, and it provides error bounds and backward error estimates for the solution. In addition to the normwise error bound, the code provides a maximum componentwise error bound if possible. Peep the comments for ERR_BNDS_NORM and ERR_BNDS_COMP for the details of those error bounds. 💖

The original system of linear equations may have been equilibrated before calling this routine, as described by arguments EQUED and S below. In this case, the solution and error bounds returned are for the original unequilibrated system. No cap.

Some optional parameters are bundled in the PARAMS array. These settings determine how refinement is performed, but often the defaults are acceptable. If the defaults are acceptable, users can pass NPARAMS = 0, which prevents the source code from accessing the PARAMS argument. Slay. 🎀

🌸 Parameters 🌸

  • UPLO — CHARACTER*1. It tells you which triangle of A is stored:
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • EQUED — CHARACTER*1. Specifies the form of equilibration that was done to A before calling this routine. This is needed to compute the solution and error bounds correctly.
    • = 'N': No equilibration
    • = 'Y': Both row and column equilibration, i.e., A has been replaced by diag(S) * A * diag(S). The right hand side B has been changed accordingly.
  • N — INTEGER. The order of the matrix A. N >= 0.
  • NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
  • A — COMPLEX array, dimension (LDA,N). The Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • AF — COMPLEX array, dimension (LDAF,N). The triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, as computed by CPOTRF.
  • LDAF — INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
  • S — REAL array, dimension (N). The scale factors for A. If EQUED = 'Y', A is multiplied on the left and right by diag(S). S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive. If S is output, each element of S is a power of the radix. If S is input, each element of S should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable. 🫶
  • B — COMPLEX array, dimension (LDB,NRHS). The right hand side matrix B.
  • LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • X — COMPLEX array, dimension (LDX,NRHS). On entry, the solution matrix X, as computed by SGETRS. On exit, the improved solution matrix X.
  • LDX — INTEGER. The leading dimension of the array X. LDX >= max(1,N).
  • RCOND — REAL. Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill-conditioned.
  • BERR — REAL array, dimension (NRHS). Componentwise relative backward error. This is the componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).
  • N_ERR_BNDS — INTEGER. Number of error bounds to return for each right hand side and each type (normwise or componentwise). See ERR_BNDS_NORM and ERR_BNDS_COMP below.

🌈 ERR_BNDS_NORM 🌈

ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows:

     Normwise relative error in the ith solution vector:
             max_j (abs(XTRUE(j,i) - X(j,i)))
            ------------------------------
                  max_j abs(X(j,i))

The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned. 👑

The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side.

The second index in ERR_BNDS_NORM(:,err) contains the following three fields:

  • err = 1 — boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
  • err = 2 — error bound: The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
  • err = 3 — Reciprocal condition number: Estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is . These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra cautions. 💜

🍓 ERR_BNDS_COMP 🍓

…and that's exactly where this part of the page cuts off, bestie — ERR_BNDS_COMP continues in part 2. 💖

💅 ERR_BNDS_COMP

ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS) — ok so for every right-hand side, this array is holding all the tea about error bounds and condition numbers for the componentwise relative error, which is defined like this:

             abs(XTRUE(j,i) - X(j,i))
        max_j ----------------------
                  abs(X(j,i))

That's the componentwise relative error in the *i*th solution vector, periodt.

The array is indexed by the right-hand side i (because that's literally what the componentwise relative error depends on), and by the type of error information, which we're getting to. Right now there are up to three pieces of info returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed at all, bestie. And if N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned. Slay.

The first index in ERR_BNDS_COMP(i,:) corresponds to the *i*th right-hand side 💫

The second index in ERR_BNDS_COMP(:,err) contains these three fields:

  • err = 1 — boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
  • err = 2 — error bound: the estimated forward error, almost certainly within a factor of 10 of the true error, so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. (Read the fine print, babe 😌)
  • err = 3 — Reciprocal condition number: the estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is . These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z.

Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1. Iconic scaling behaviour, no cap ✨

ok the source literally cuts off mid-sentence there ("the error estimate is .") — not me, I'm just the messenger 😭💀

See Lapack Working Note 165 for further details and extra cautions. iykyk 📖

🔢 NPARAMS

NPARAMS is INTEGER — specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used. Simple as that 🎀

🎛️ PARAMS

PARAMS is REAL array, dimension NPARAMS — specifies algorithm parameters. If an entry is < 0.0, then that entry gets filled with the default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters. Very demure, very mindful 🌸

  • PARAMS(LA_LINRX_ITREF_I = 1) — whether to perform iterative refinement or not.
    • Default: 1.0
    • = 0.0: No refinement is performed, and no error bounds are computed.
    • = 1.0: Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION.
    • (other values are reserved for future use)
  • PARAMS(LA_LINRX_ITHRESH_I = 2) — maximum number of residual computations allowed for refinement.
    • Default: 10
    • Aggressive: set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in err_bnds_norm and err_bnds_comp may no longer be trustworthy. ⚠️ so like, bestie, be careful with that one.
  • PARAMS(LA_LINRX_CWISE_I = 3) — flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false.
    • Default: 1.0 (attempt componentwise convergence) 🫶

🧰 WORK

WORK is COMPLEX array, dimension (2*N) — your lil scratch space, go off 💜

🌱 RWORK

RWORK is REAL array, dimension (2*N) — the real-valued scratch space, keeping it grounded 🌈

🚦 INFO

INFO is INTEGER — the vibe check at the end:

  • = 0: Successful exit. The solution to every right-hand side is guaranteed. We love a clean run 👑
  • < 0: If INFO = -i, the *i*th argument had an illegal value 🙈
  • > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. 💀 rip.
  • = N+J: The solution corresponding to the Jth right-hand side is not guaranteed. The solutions corresponding to other right-hand sides K with K > J may not be guaranteed as well, but only the first such right-hand side is reported.

If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth right-hand side is the first with a normwise error bound that is not guaranteed (the smallest J such that ERR_BNDS_NORM(J,1) = 0.0).

By default (PARAMS(3) = 1.0) the Jth right-hand side is the first with either a normwise or componentwise error bound that is not guaranteed (the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0). See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1) for the deets.

To get information about all of the right-hand sides, check ERR_BNDS_NORM or ERR_BNDS_COMP. Don't be lazy, read the whole array 🔥

🌟 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 389 of file cporfsx.f. ✨

🦋 Author (yes, again)

Generated automatically by Doxygen for LAPACK from the source code. The robots wrote this one, bestie 🤖💖

aaaand that's a wrap on part 2, babe 🎀 huge thank you to the LAPACK girlies — LAPACK Version 3.12.0, page cporfsx.f(3) — go read the source, it's giving main character energy 🌈👑 mwah 💋
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